Relative Fulfilment in Poly Relationship Networks

[ashley_y]

Love is finite: no matter how you feel, there are only so many hours in the day.

It seems to me in polyamorous relationships, those involved in more relationships are in some sense better fulfilled, while those involved with them are less so. Consider: A and B are in a relationship. If B starts a relationship with C, B gains a new relationship, while A loses some of B's attention.

But by just how much? Can we model these effects for general relationship arrangements?
Graph

Let's make some assumptions:

1. Relationships are binary: that is, between two people.
2. Relationships are symmetric, that is, reciprocated.
3. The intensity of a relationship is determined only by the structure of the relationship network. To keep things simple, at least at first, no "primaries" and "secondaries".

Now we can represent polyamory using graph theory. A network of poly relationships is an undirected graph, where the vertices are people and the edges are relationships. You can draw this as you might expect, with dots for people and lines for relationships. The graph is undirected because relationships are symmetric, so you don't draw arrowheads on the lines.

Now what we want to do is calculate how much each person is fulfilled by their relationships. The strategy is to evaluate an "intensity" of some kind for each relationship, and then combine these somehow to calculate fulfilment. We're just attaching numbers to the dots and lines. Some more assumptions:

1. Adding relationships increases a person's fulfilment.
2. Adding relationships decreases the intensity of existing relationships.

For the combining function, we'll use summation, just because it's easiest. Thus one's fulfilment is the sum of intensities of all the relationships one is in.

What about intensity? Let's consider a simple example to guide intuition. Imagine a single "hub" person in n relationships each with people in no other relationships. From assumption 3 above, all relationships have the same intensity, which we'll call r, so the hub person's fulfilment is nr. Assumptions 4 and 5 give us this:

• As n increases, nr increases.
• As n increases, r decreases.

To satisfy this r has to be "between" k/n and k: the simplest model is r = 1/√n. This suggests a general function for the intensity of a relationship: 1/√(pq), where p and q are the number of relationships the two are in.
Model

So, to summarise:

• The engagement of a person is the number of relationships they are in ("degree of the vertex" in graph theory).
• The intensity of a relationship is the inverse geometric mean (i.e. 1/√(pq)) of the engagements of its two participants.
• The fulfilment of a person is the sum of the intensities of all the relationships they are in.

Let's consider some examples:

• 
  Two people are in a relationship: a common arrangement known as a "polyfidelitous dyad" (not really). The engagement of each is 1, the intensity of the relationship is 1, the fulfilment of each person is 1. That's a nice straightforward baseline.
  
• 
  A "V" relationship: B is in a relationship with A and C. B has engagement 2, A and C have engagement 1. Each relationship has intensity 1/√2. A and C have fulfilment of 1/√2 ≈ 0.7, while B has a fulfilment of √2 ≈ 1.4.
  
• 
  "Mixed doubles", two men and two women in heterosexual relationships, forming a square. The engagement of each is 2, the intensity of each relationship is 1/2, and the fulfilment of each is 1.
  

This last points out a more general result: if everyone has the same (non-zero) engagement, then everyone has a fulfilment of 1. This result has an appealing neutrality if one bears in mind that we are not considering personal preference towards polyamory. In practice, one might imagine that those who prefer polyamory will have a higher fulfilment (due to more relationships) with a higher common engagement, while those who do not will have a lower fulfilment (due to the lower intensity of each relationship); but we don't include this in the model.

Note also that a relationship can never have an intensity greater than 1, and only monogamous "full-time" relationships reach that.
Weight

How might we model "primary" and "secondary" arrangements? One simple way is to draw two relationship lines for a primary relationship, and one for a secondary. Our graphs now become multigraphs. This is appealing in that multiplying everyone's relationships doesn't change anything: that is, if we take a relationship graph and double all the edges, the engagements double, the intensity of the relationships is simply evenly split, and the fulfilments remain the same.

One can generalise this by assigning a weight to each relationship. Now the engagement of a person is the sum of weights, and the intensity of a relationship with weight w becomes w/√(pq). Again scaling weights across a whole graph doesn't affect any intensities or fulfilments, and again relationship intensities can never be more than 1.

For example, A and B consider themselves in a "primary" relationship, which we'll give weight 2. B and C are in a "secondary" relationship of weight 1. Engagements for A, B and C are 2, 3, 1. The A-B relationship has intensity √(2/3) and the B-C relationship has intensity 1/2. Fulfilments are approximately 0.8, 1.3, 0.5 (as opposed to the unweighted fulfilments calculated above, 0.7, 1.4, 0.7). Note how A has gained from this weighting, while B and (especially) C have lost.
The Second Order

One shortcoming of our model is that it doesn't account for second-order effects. For instance, let us imagine that A is in a relationship with B, who is also in a relationship with C, who is in n relationships. In our model, as n increases, the B-C relationship becomes less intense, as one would expect. But the A-B intensity is unaffected, it's always 1/√2. Shouldn't the A-B relationship become more intense as B gets less time to spend with C?

One tempting approach is to iterate the process, substituting fulfilment for engagement. This scheme does propagate higher-order effects appropriately, but annoyingly it does not in general converge (consider the mixed doubles square). There may however be self-consistent solutions, let me know if you find a general method for calculating them.